Unit 5 of Pre-Calculus: Analytic Trigonometry.
This unit covers verifying identities, sum and difference formulas and double-angle formulas — essential concepts for Pre-Calculus. Use our interactive study games to test your understanding, or review questions in traditional format below.
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This unit covers verifying identities, sum and difference formulas and double-angle formulas — essential concepts for Pre-Calculus. Use our interactive study games to test your understanding, or review questions in traditional format below.
Key Concepts Breakdown
1 Verifying Identities
Students must be able to prove that a trigonometric equation is an identity by transforming one side until it matches the other. Work on only one side at a time — never move terms across the equal sign. Mastery of the Pythagorean, reciprocal, and quotient identities is required.
Key Points
- Start with the more complex side and simplify toward the simpler side
- Key identities: sin²x + cos²x = 1, tan x = sin x/cos x, sec x = 1/cos x, csc x = 1/sin x, cot x = cos x/sin x
- Common strategies: factor, convert everything to sin/cos, multiply by a conjugate, or split a fraction
- You cannot assume the identity is true — never cross-multiply or add to both sides
Verify: (1 - cos²x) / sin x = sin x
Start with the left side: replace 1 - cos²x with sin²x using the Pythagorean identity, giving sin²x / sin x. Cancel one factor of sin x to get sin x, which matches the right side exactly. The identity is verified.
2 Sum and Difference Formulas
Students must know the formulas for sin(A ± B), cos(A ± B), and tan(A ± B) and be able to apply them to find exact values of non-standard angles. These formulas are also used to simplify expressions and verify identities involving compound angles.
Key Points
- sin(A ± B) = sin A cos B ± cos A sin B
- cos(A ± B) = cos A cos B ∓ sin A sin B (note: signs are OPPOSITE for cosine)
- tan(A ± B) = (tan A ± tan B) / (1 ∓ tan A tan B)
- Use reference angles like 30°, 45°, 60° to find exact values — for example, 75° = 45° + 30°
Find the exact value of cos(75°).
Write 75° as 45° + 30° and apply the cosine sum formula: cos(45°)cos(30°) - sin(45°)sin(30°). Substitute exact values: (√2/2)(√3/2) - (√2/2)(1/2) = √6/4 - √2/4. The exact value is (√6 - √2)/4.
3 Double-Angle Formulas
Students must know all three forms of the cosine double-angle formula and be able to choose the correct form based on what information is given. These formulas are used to find exact values, simplify expressions, and solve equations.
Key Points
- sin(2x) = 2 sin x cos x
- cos(2x) has three forms: cos²x - sin²x, 2cos²x - 1, or 1 - 2sin²x — choose the form that fits the given information
- tan(2x) = 2 tan x / (1 - tan²x)
- If given sin x or cos x and a quadrant, find the missing value using sin²x + cos²x = 1 before applying the formula
If sin x = 3/5 and x is in Quadrant I, find sin(2x) and cos(2x).
Since sin x = 3/5 in QI, use the Pythagorean identity to find cos x = 4/5. Apply the double-angle formulas: sin(2x) = 2(3/5)(4/5) = 24/25. For cos(2x) use cos²x - sin²x = (4/5)² - (3/5)² = 16/25 - 9/25 = 7/25.
Questions, answered.
What is Analytic Trigonometry?
Analytic Trigonometry is Unit 5 of Pre-Calculus, covering verifying identities, sum and difference formulas and double-angle formulas.
How to study for Pre-Calculus Unit 5?
Start with the Quick Summary above, review the Key Concepts, then test yourself with our interactive study games. Aim for 80%+ accuracy before moving on.
How many questions are in this unit?
This unit has 28+ review questions across 5 different game modes.